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1. if A is a matrix such that A² = 0, then A is not invertible.

1. if A is a matrix such that A² = 0, then A is not invertible.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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True or False? Prove.

**Matrix Invertibility Condition** 1. If \( A \) is a matrix such that \( A^2 = 0 \), then \( A \) is not invertible. Explanation: For a matrix \( A \) to be invertible, there must exist a matrix \( B \) such that \( AB = BA = I \), where \( I \) is the identity matrix. If \( A^2 = 0 \), then multiplying both sides of this equation by \( A^{-1} \) (assuming \( A \) is invertible) leads to a contradiction. Therefore, \( A \) cannot have an inverse, indicating that \( A \) is not invertible. This is a fundamental property in linear algebra used to determine the invertibility of matrices.
Transcribed Image Text:**Matrix Invertibility Condition** 1. If \( A \) is a matrix such that \( A^2 = 0 \), then \( A \) is not invertible. Explanation: For a matrix \( A \) to be invertible, there must exist a matrix \( B \) such that \( AB = BA = I \), where \( I \) is the identity matrix. If \( A^2 = 0 \), then multiplying both sides of this equation by \( A^{-1} \) (assuming \( A \) is invertible) leads to a contradiction. Therefore, \( A \) cannot have an inverse, indicating that \( A \) is not invertible. This is a fundamental property in linear algebra used to determine the invertibility of matrices.
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